Count Splitting Controls For Type 1 Error in Differential Testing After Tree Merging of Gene Isoforms
Justin Landis1, Michael Love2, 3
1 Biological & Biomedical Sciences Program
2 Department of Genetics, UNC Chapel Hill
3 Department of Biostatistics, UNC Chapel Hill
Background
One Gene may be translated into mRNA and then spliced into multiple transcripts named isoforms. Isoforms may be modulated by splicing factors within the cell.1
RNAseq takes a snapshot of a cell’s gene expression profile at the time of sequencing. Short read sequencing, 100-250 base pairs, required estimating which isoform is present since the read may be contained within one exon. Long read sequencing may span multiple exons, providing more confidence on the transcript isoform detected at the expense of read depth.
Goal: Develop an isoform-grouping method to facilitate isoform-level differential expression (DE) analysis using long-read sequencing data while controlling for False Positives for differential expression.
DE Testing of Inner Nodes
A gene with \(N\) isoforms implies \(N-1\) inner nodes for its associated tree. These inner nodes are the sum of the leaves of a sample.
Once we have our extended data, we perform DESeq22 and evaluate the resulting pvalues with our tree Climbing algorithm.
Cluster Tree Generation Method
To Generate Hierarchical clusters, we generated a similarity metric based on the similarities between transcripts as opposed to using data dependent counts. Let \(G\) represent a set of isoforms of size \(g\). given any indexes \(i\), \(j \le g\) , define \(G_i\) and \(G_j\) as isoforms \(i\) and \(j\) from \(G\) such that they represent sets of exons of size \(N\) and \(M\) respectively. For any two \(i\) and \(j\), we can define the similarity as:
\[ S_{ij}(G_i,G_j) = \frac{2\sum_n^N\sum_m^MJ(G_{i_n}, G_{j_m})}{N + M} \hspace{1cm} J(G_{i_n}, G_{j_m}) = \frac{G_{i_n}\cap G_{j_m}}{G_{i_n}\cup G_{j_m} } \]
Simulation Methods
Choose an inner node within the tree and shift the mean of all leaves for a particular group by some delta. We can evaluate our Tree climbing algorithm based on how well it accurately chooses the known perturbed nodes.
Let \(\mu_{ij}=\mu_0 + \delta_{ij}\), \(\mu_0 = 10\) and \(\delta_{ij} = 0\) for \(i,j\) in the control group, and \(\delta_{ij} = \hat \delta\) for \(i,j\) in the affected group. Each entry, \(X_{ij}\), is sampled as follows. \[ X_{ij} \sim \text{Nbinom}(\mu_{ij},\ \alpha = 100) \]
Simulation Sensitivity
We conducted 50 simulations per \(\hat\delta \in \{0.5, 1, 1.5, ..., 6\}\) and evaluated how often our tree climbing algorithm correctly merged the known perturbed data. Merged nodes with \(\delta_{ij}=0\) are False Positives and unmerged nodes of with \(\delta_{ij}=\hat\delta\) are considered False Negatives in the tree climbing context.
Count Splitting3
While at higher \(\hat\delta\) shifts in our simulations are able to identify the correct inner node, but we also need to consider the case of the null hypothesis. To assess the distribution of pvalues under the assumption that there is no difference in the \(\mu_{ij}\), we ran simulations with \(\hat\delta = 0\).
There is an enrichment of low pvalues among the inner nodes in the null hypothesis simulations. To correct this we can apply count splitting. The count splitting method is formulated for experiments that use the same data for feature selection as they use for analysis.3
\[ X_{ij} \sim \text{Nbinom}(\mu_{ij}, \alpha = 100)\\ X_{ij}^\text{train} \sim \text{Bin}(X_{ij}, \theta = 0.5)\\ X_{ij}^\text{test} = X_{ij} - X_{ij}^\text{train} \]
Conclusions
- Simulations of a single gene of 44 isoforms can reliably detect the correct node with \(\log_2\) fold change of 0.5 between groups.
- Utilization of count splitting controls for type 1 error under the null hypothesis.
Future Directions
- Improve speed of data merging step.
- Apply tree climbing and count splitting methods on real data sets.